2 Objectives Write the standard equation for a hyperbola. Graph a hyperbola, and identify its center, vertices, co-vertices, foci, and asymptotes.
3 HyperbolaWhat would happen if you pulled the two foci of an ellipse so far apart that they moved outside the ellipse?The result would be a hyperbola, another conic section.A hyperbola is a set of points P(x, y) in a plane such that the difference of the distances from P to fixed points F1 and F2, the foci, is constant. For a hyperbola, d = |PF1 – PF2 |, where d is the constant difference. You can use the distance formula to find the equation of a hyperbola.
4 Example 1: Using the Distance Formula to Find the Constant Difference of a Hyperbola Find the constant difference for a hyperbola with foci F1 (–8, 0) and F2 (8, 0) and the point on the hyperbola (8, 30).Sol:d = |PF1 – PF2 | Definition of the constant difference of a hyperbola.Distance Formula
5 solutionSubstituted = 4The constant difference is 4.
6 Check it out! ExampleFind the constant difference for a hyperbola with foci F1 (0, –10) and F2 (0, 10) and the point on the hyperbola (6, 7.5).Sol:The constant difference is 12.
7 HyperbolasAs the graphs in the following table show, a hyperbola contains two symmetrical parts called branches.A hyperbola also has two axes of symmetry. The transverse axis of symmetry contains the vertices and, if it were extended, the foci of the hyperbola. The vertices of a hyperbola are the endpoints of the transverse axis.The conjugate axis of symmetry separates the two branches of the hyperbola. The co-vertices of a hyperbola are the endpoints of the conjugate axis. The transverse axis is not always longer than the conjugate axis.
8 HyperbolasThe standard form of the equation of a hyperbola depends on whether the hyperbola’s transverse axis is horizontal or vertical.
9 HyperbolasThe values a, b, and c, are related by the equation c2 = a2 + b2. Also note that the length of the transverse axis is 2a and the length of the conjugate is 2b.
10 Example 2A: Writing Equations of Hyperbolas Write an equation in standard form for each hyperbola.Sol:Step 1 Identify the form of the equation.The graph opens horizontally, so the equation will be in the form ofx2a2– = 1y2b2
11 solution Step 2 Identify the center and the vertices. The center of the graph is (0, 0), and the vertices are (–6, 0) and (6, 0), and the co-vertices are (0, –6), and (0, 6). So a = 6, and b = 6.Step 3 Write the equation.x236– = 1.y2Because a = 6 and b = 6, the equation of the graph is62– = 1, or
12 Example 2B: Writing Equations of Hyperbolas Write an equation in standard form for each hyperbola.The hyperbola with center at the origin, vertex (4, 0), and focus (10, 0).Sol:Step 1 Because the vertex and the focus are on the horizontal axis, the transverse axis is horizontal and the equation is in the formx2a2– = 1y2b2
13 solutionStep 2 Use a = 4 and c = 10; Use c2 = a2 + b2 to solve for b2.102 = 42 + b284 = b2Step 3 The equation of the hyperbola isx216– = 1y284
14 Check it out! ExampleWrite an equation in standard form for each hyperbola.Vertex (0, 9), co-vertex (7, 0)Sol:Because a = 9 and b = 7, the equation of the graph is , ory292– = 1x2728149
15 HyperbolasAs with circles and ellipses, hyperbolas do not have to be centered at the origin.
16 Example 3A: Graphing a Hyperbola Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph.Sol:x249– = 1y29Step 1 The equation is in the form so the transverse axis is horizontal with center (0, 0).x2a2– = 1y2b2
17 solutionStep 2 Because a = 7 and b = 3, the vertices are (–7, 0) and (7, 0) and the co-vertices are (0, –3) and (0, 3).37Step 3 The equations of the asymptotes arey = x and y = – x.
18 solutionStep 4 Draw a box by using the vertices and co-vertices. Draw the asymptotes through the corners of the boxStep 5 Draw the hyperbola by using the vertices and the asymptotes.
19 Example 3B: Graphing a Hyperbola Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph.(x – 3)29– = 1(y + 5)249
20 ParameterNotice that as the parameters change, the graph of the hyperbola is transformed.